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The thesis studies Frobenius-type theorems in non-smooth settings.  We extend the definition of involutivity to non-Lipschitz subbundles usinggeneralized functions. We prove the real Frobenius Theorem with sharpregularity on log-Lipschitz subbundles. We also develop a singular version ofthe Frobenius theorem on log-Lipschitz vector fields: if $X_1,\dots,X_m$ arelog-Lipschitz vector fields such that $[X_i,X_j]=\sum_{k=1}^mc_{ij}^kX_k$ where$c_{ij}^k$ are the derivatives of log-Lipschitz functions, then for any point$p$ there is a $C^1$-manifold containing $p$ such that $X_1,\dots,X_m$ span itstangent space. On the quantitative side, if $c_{ij}^k\in C^{\alpha-1}$ where$1<\alpha<2$ then on each leaf where $X_1,\dots,X_m$ span the tangent spaces wecan find a regular parameterization $\Phi$ such that$\Phi^*X_1,\dots,\Phi^*X_m$ are $C^\alpha$, and their $C^\alpha$ norm dependonly on the diffeomorphic invariant quantities of $X_1,\dots,X_m$.  For a complex Frobenius structure there is a coordinate chart $F$ that takesimage in $\mathbb R^r_t\times\mathbb C^m_z\times \mathbb R^{N-r-2m}_s$, suchthat the structure is locally spanned by $F^*\partial_t,F^*\partial_z$. When ithas H\"older regularity $\alpha>1$, we show that the coordinate chart $F$ maybe taken to be $\mathscr C^\alpha$, and the vector fields$F^*\partial_t,F^*\partial_z$ are $\mathscr C^{\alpha-\epsilon}$ for every$\epsilon>0$. We give an example to show that the regularity result for$F^*\partial_z$ is optimal. When a complex Frobenius structure $S$ is$C^\alpha$ ($\frac12<\alpha\le1$) such that $S+\bar S$ is log-Lipschitz, thenfor every $\epsilon>0$ there is a $C^{2\alpha-1-\epsilon}$ homeomorphism$\Phi(t,z,s)$ such that $S$ is spanned by $\Phi_*\partial_t,\Phi_*\partial_z\inC^{2\alpha-1-\epsilon}$.

On Rough Frobenius-type Theorems and Their H\"older Estimates